Lazy Migration Reordering Algorithms

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 p 5000 10000 15000 20000 25000 30000 35000 40000 l 2 4 6 8 10 12 log2β0 l,p+1 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 p 5000 10000 15000 20000 25000 30000 35000 40000 l 1 2 3 4 5 6 7 8 9 10 log2β1 l,p+1 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 p 5000 10000 15000 20000 25000 30000 35000 40000 l 1 2 3 4 5 6 7 log2β2 l,p +1 Fig. 8.8. Graphs of log 2 β l ,p k + 1 for k = 0, 1, 2, respectively, for zeolite FAU. The graphs are sampled onto an 80 by 80 grid. subtractive, starting from white and ending with black. We use shades of cyan, magenta, and yellow for representing values of β , β 1 , and β 2 , respectively. Given this system, we can now visualize the complete topological content of a space in a single image. We call these images topology maps. Given a topol- ogy map, we can immediately observe the salient topological features of the associated space. Example 8.1 topology map of FAU Figure 8.10 displays the topology maps of FAU, corresponding to its Betti and square Betti numbers. The map of FAU has six regions, clearly delineated by color. There is a seventh dark cyan region in the top left corner, describing the arrival of all the vertices. We per- ceive that persistent components are formed in the large cyan triangle: The vertices arrive, are connected into structures with tunnels creating 1-cycles in the blue triangle, completed into voids creating 2-cycles in the green region, and finally filled up with tetrahedra. In the second stage, these components are connected to form a single structure with tunnels magenta triangle and form voids again yellow triangle, which are again filled. Each point of a topology map l, p corresponds to a p-persistent complex K l ,p . Consequently, these maps provide us with an powerful navigational tool for software design. I use these maps in my topology visualization program, CView , which I will describe in Chapter 11. We end this chapter with a few visualizations of persistent complexes. We claimed earlier that topology maps were useful for displaying the entire topo- logical content of a space. We substantiate these structural predictions in Ex- ample 8.1 by showing persistent complexes from the various regions of the topology map of FAU in Figure 8.11. The Betti numbers of the complexes are listed underneath them. We may also use the persistent algorithm to view cy- cles and their manifolds in each complex. Figure 8.12 displays the eight voids of a persistent complex for zeolite KFI. We will see more cycles and manifolds in Chapter 10, when discussing the linking number algorithm.